Discrete Structures: For each of the following relations, determine whether it i

Question

Discrete Structures:

For each of the following relations, determine whether it is reflexive, anti-reflexive, symmetric, anti-symmetric, or transitive. Note that it may be possible for a relation to satisfy more than one property in each category – think carefully about the definitions! Briefly explain your answers for each one.

c. The domain is all real numbers. For any real numbers x and y, xRy if |x – y| 2.

d. The domain is all real numbers. For any real numbers x and y, xRy if x + y = 0.

e. The domain is all sloths. For any sloths x and y, xRy if x and y are the same sloth.

Explanation / Answer

Reflexive: A relation R on a set A is called reflexive if (a, a) R for every element a A.

Anti-Reflexive: A relation R on a set A is called anti reflexive if (a, a) ! R (does not belong to R) for every element a A.

Symmetric: A relation R on a set A is called symmetric if (b, a) R whenever (a, b) R, for all a, b A.

Antisymmetric: A relation R on a set A such that for all a, b A, if (a, b) R and (b, a) R, then a = b is called antisymmetric.

Transitive: A relation R on a set A is called transitive if whenever (a, b) R and (b, c) R, then (a, c) R, for all a, b, c A

c. The domain is all real numbers. For any real numbers x and y, xRy if |x – y| 2 :

          i) f(2,2) =|2-2|=0 and 0<=2 hence f(a,a) accepted for a=2 ,hence reflexive.

ii) it is reflexive hence it cant be anti reflexive. because it accepts f(5,5)<=0 ...so not anti reflexive.

   iii) |a-b|=|b-a| hence f(x,y) = f(y,x) hence both f(x,y) and f(y,x) are accepted,hence it is symmetric.

  iv) f(2,-2)=f(-2,2) ; f(x,y) and f(y,x) both are accepted but x != y,hence it is not antisymmetric.

v) f(2,3)=1 <= 2 and f(3,5)=2 <=2 but f(2,5)=3 >2; ; f(x,y) and f(y,z) are accepted but f(x,z) not accepted,henceit is not reflexive.

d.The domain is all real numbers. For any real numbers x and y, xRy if x + y = 0 :

it is a set of (a,-a) elements

          i) f(2,2) =2+2=4 and 4!=0 hence f(a,a) not accepted for a!=0 ,hence not reflexive.

ii) for value 0,f(0,0)=0+0=0 it accepts f(a,a) for a=0 hence it is not anti reflexive.

   iii) a+b=b+a hence f(x,y) = f(y,x) hence both f(x,y) and f(y,x) are accepted,hence it is symmetric.

  iv) f(2,-2)=f(-2,2) ; f(x,y) and f(y,x) both are accepted but x != y,hence it is not antisymmetric.

v) f(2,-2)=0 and f(-2,2)=0 but f(2,2)=4 and 4!=0 ; f(x,y) and f(y,z) are accepted but f(x,z) not accepted,henceit is not reflexive.

e. The domain is all sloths. For any sloths x and y, xRy if x and y are the same sloth :

it is a set of (sloth_a,sloth_a) elements ..i.e (a,a)

          i) f(sloth_a,sloth_a) is accepted because two are same sloth,hence reflexive.

ii) from above it accepts f(sloth_a,sloth_a) hence it is not anti reflexive.

   iii) we are having only set of (x,x) elements,hence it is not symmetric.

  iv) we are having only set of (x,x) elements,hence it is not antisymmetric.

v) we are having only set of (x,x) elements,henceit is not reflexive.

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